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A Generalization of Normal Spaces ARCHIVUM MATHEMATICUM (BRNO) Tomus 44 (2008), 265–270 A GENERALIZATION OF NORMAL SPACES V. Renukadevi and D. Sivaraj Abstract. A new class of spaces which contains the class of all normal spaces is defined and its characterization and properties are discussed. 1. Introduction and preliminaries Ideals in topological spaces have been considered since 1930. This topic has won its importance by the paper of Vaidyanathaswamy [8]. An ideal I on a topological space (X, τ) is a nonempty collection of subsets of X which satisfies (i) A ∈ I and B ⊂ A implies B ∈ I and (ii) A ∈ I and B ∈ I implies A ∪ B ∈ I. Given a topological space (X, τ) with an ideal I on X and if ℘(X) is the set of all subsets of X, a set operator (·)? : ℘(X) → ℘(X), called the local function [4] of A with respect to τ and I, is defined as follows: for A ⊂ X, A?(I, τ) = x ∈ X | U ∩A 6∈ I for every U ∈ τ(x) where τ(x) = U ∈ τ | x ∈ U . A Kuratowski closure operator cl?(·) for a topology τ ?(I, τ), called the ?-topology, finer than τ is defined by cl?(A) = A ∪ A?(I, τ) [9]. When there is no chance for confusion, we will simply write A? for A?(I, τ). We will make use of the properties of the local function established in Theorem 2.3 of [3] without mentioning it explicitly. The aim of this paper is to introduce a new class of spaces called I-normal spaces which contains the class of all normal spaces and discuss some of its properties. By a space, we always mean a topological space (X, τ) with no separation properties assumed. If A ⊂ X, cl(A) and int(A) will, respectively, denote the closure and interior of A in (X, τ). An ideal I is said to be codense or a boundary ideal [5] if τ ∩ I = {∅}. The following lemmas will be useful in the sequel. Lemma 1.1 ([6, Theorem 5]). Let (X, τ, I) be an ideal space and A be a subset of X. If A ⊂ A?, then A? = cl(A?) = cl(A) = cl?(A). Lemma 1.2 ([3, Theorem 6.1]). If (X, τ, I) is an ideal space, then I is codense if and only if G ⊂ G? for every open set G in X. 2. I-normal spaces An ideal space (X, τ, I) is said to be I-normal if for every pair of disjoint closed sets A and B of X, there exist disjoint open sets U and V such that A − U ∈ I and 2000 Mathematics Subject Classification: primary 54D15; secondary 54C10. Key words and phrases: I-regular, codense ideal, I-compact, I-paracompact. Received December 4, 2006. Editor A. Pultr. 266 V. RENUKADEVI AND D. SIVARAJ B − V ∈ I. Clearly, if I = {∅}, then normality and I−normality coincide. Also, if I and J are ideals on X such that I ⊂ J and (X, τ, I) is I-normal, then (X, τ, J ) is J -normal. Since ∅ ∈ I, it is clear that every normal space is an I-normal space for every ideal I but not the converse, as shown by the following Example 2.1. Example 2.1. Consider the Modified Fort space [7, Example 27] in which X = N ∪ {x1} ∪ {x2}, where N is the set of all natural numbers, with the topology τ defined as follows: Any subset of N is open and any set containing x1 or x2 is open if and only if it contains all but a finite number of points of N. This space is not normal. Consider If , the ideal of all finite subsets of X. We prove that (X, τ, If ) is I-normal. Let A and B be two disjoint closed sets in X. Case (i). If A and B are subsets of N, then A and B are open. If G = A and H = B, since ∅ ∈ I, A − G ∈ If and B − H ∈ If . Case (ii). Suppose x1 ∈ A and x2 6∈ A. Let G = A−{x1} and H = (X −A)−{x2}. Then G and H are disjoint. Since G ⊂ N, G is open and A − G = {x1} ∈ If . Since H ⊂ N, H is open and B − H ⊂ B ∩ A ⊂ A. Since x2 6∈ A, A is finite and so A ∈ If which implies that B − H ∈ If . Thus, there exist disjoint open sets G and H such that A − G ∈ If and B − H ∈ If . Case (iii). Suppose x1, x2 ∈ A. Let G = A − {x1, x2} and H = B. Then G and H are disjoint. Since G ⊂ N, G is open and A − G = {x1, x2} ∈ If . x1, x2 6∈ B implies that B ⊂ N and so B is open. Thus there exist disjoint open sets G and H such that A − G ∈ If and B − H ∈ If . Thus, in all the three cases, there exist disjoint open sets G and H such that A − G ∈ If and B − H ∈ If . Hence (X, τ, If ) is I-normal. The following Theorem 2.2 characterizes I-normal spaces. Theorem 2.2. Let (X, τ, I) be an ideal space. Then the following are equivalent. (a) (X, τ, I) is I-normal. (b) For every closed set F and open set G containing F , there exists an open set V such that F − V ∈ I and cl(V ) − G ∈ I. (c) For each pair of disjoint closed sets A and B, there exists an open set U such that A − U ∈ I and cl(U) ∩ B ∈ I. Proof. (a)⇒(b). Let F be closed and G be open such that F ⊂ G. Then X − G is a closed set such that (X − G) ∩ F = ∅. By hypothesis, there exist disjoint open sets U and V such that (X − G) − U ∈ I and F − V ∈ I. Now U ∩ V = ∅ implies that cl(V ) ⊂ X − U and so (X − G) ∩ cl(V ) ⊂ (X − G) ∩ (X − U) which in turn implies that cl(V ) − G ⊂ (X − G) − U ∈ I. Therefore, cl(V ) − G ∈ I. (b)⇒(c). Let A and B be disjoint closed subsets of X. Then there exists an open set U such that A − U ∈ I and cl(U) − (X − B) ∈ I which implies that A − U ∈ I and cl(U) ∩ B ∈ I. (c)⇒(a). Let A and B be disjoint closed subsets in X. Then there exists an open set U such that A − U ∈ I and cl(U) ∩ B ∈ I. Now cl(U) ∩ B ∈ I implies that B − (X − cl(U)) ∈ I. If V = X − cl(U), then V is an open set such that B − V ∈ I and U ∩ V = U ∩ (X − cl(U)) = ∅. Hence (X, τ, I) is I-normal. A GENERALIZATION OF NORMAL SPACES 267 The following Corollary 2.3 follows from Theorem 2.2 and Lemmas 1.1 and 1.2. Corollary 2.3. Let (X, τ, I) be an ideal space where I be codense. Then the following are equivalent. (a) (X, τ, I) is I-normal. (b) For every closed set F and open set G containing F , there exists an open set V such that F − V ∈ I and V ? − G ∈ I. (c) For each pair of disjoint closed sets A and B, there exists an open set U such that A − U ∈ I and U ? ∩ B ∈ I. If I is an ideal of subsets of X and Y is a subset of X, then IY = {Y ∩ I | I ∈ I} = {I ∈ I | I ⊂ Y } is an ideal of subsets of Y [5]. The following Theorem 2.4 shows that I-normality is closed hereditary. Since every space (X, τ) is the ideal space (X, τ, I) where I = {∅}, it follows that the condition closed on the subset cannot be dropped. Theorem 2.4. If (X, τ, I) is an I-normal ideal space and Y ⊂ X is closed, then (Y, τY , IY ) is IY -normal. Proof. Let A and B be disjoint τY closed subsets of Y . Since Y is closed, A and B are disjoint closed subsets of X. By hypothesis, there exist disjoint open sets U and V such that A − U ∈ I and B − V ∈ I. If A − U = I ∈ I and B − V = J ∈ I, then A ⊂ U ∪ I and B ⊂ V ∪ J. Since A ⊂ Y , A ⊂ Y ∩ (U ∪ I) and so A ⊂ (Y ∩ U) ∪ (Y ∩ I). Therefore, A − (Y ∩ U) ⊂ (Y ∩ I) ∈ IY . Similarly, B − (Y ∩ V ) ⊂ (Y ∩ J) ∈ IY . If U1 = Y ∩ U and V1 = Y ∩ V , then U1 and V1 are disjoint τY open sets such that A − U1 ∈ IY and B − V1 ∈ IY . Hence (Y, τY , IY ) is IY -normal. If (X, τ, I) is an ideal space, (Y, σ) is a topological space and f : (X, τ, I) → (Y, σ) is a function, then f(I) = f(I) | I ∈ I is an ideal on Y [5]. The following Theorem 2.5 shows that I-normality is a topological property. Theorem 2.5. If (X, τ, I) is an I-normal space and f : (X, τ, I) → Y, σ, f(I) is a homeomorphism, then Y, σ, f(I) is a f(I)-normal space. Proof. Let A and B be disjoint σ-closed subsets of Y . Since f is continuous, f −1(A) and f −1(B) are disjoint closed subsets of X. Since (X, τ, I) is I-normal, there exist disjoint open sets U and V in X such that f −1(A) − U ∈ I and f −1(B) − V ∈ I, f −1(A) − U ∈ I ⇒ f(f −1(A) − U) ∈ f(I) ⇒ A − f(U) ∈ f(I).
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